Embed Size (px)
Citation preview
VOLUME 14, NUMBER 20 P H Y S I C A L R E V I E W L E T T E R S 17 MAY 1965
tions). The AT ^ f rule then forbids T = 3, so the decay amplitudes a1 2 are purely symmetric, with T = 1. Our theorem can therefore by applied, and gives
2 R e ° i V ) < 1 / 5 0 , ' +\a9
If \a1\^ \a2\, then ax and a2 must be relatively imaginary to within about 1°.
We are grateful for conversations with F. S. Crawford, Jr .
* Re search supported in part by the U. S. Air Force Office of Scientific Research, Grant No. AF-AFOSR-232-63.
tAlfred P. Sloan Foundation Fellow. iJ . A. Anderson, F. S. Crawford, Jr . , R. L. Golden,
D. Stern, T. O. Binford, and V. G. Lind, Phys. Rev.
Letters 14, 475 (1965). 2T. D. Lee, R. Oehme, and C. N. Yang, Phys. Rev.
106, 340 (1957). We follow the convention of these authors in using Kj0 and2C2° to denote the long- and short-lived components, which in general are not the sameasK°±X f l .
3The authors of reference 1 have, at our suggestion, reanalyzed their data with the additional constraint Re(flt/a2) = 0. They then find Im(a1/a2) = +0.90±0.50. The corresponding intensity ratio is r 1 / r 2 = 0.8lto°Ji> giving betting odds of 10 to 1 that r 1 / r 2 is less than 2.5, and 100 to 1 that it is less than 5. The effect of the additional constraint is to reduce their upper limit on Tj/r 2 by a factor of two. They still cannot rule out r j / r 2 = 0 . (Private communication from F. S. Crawford, Jr.)
4J. H. Christensen, J. W. Cronin, V. L. Fitch, and R. Turlay, Phys. Rev. Letters 13̂ , 138 (1964).
5F. S. Crawford, J r . , private communication.
MIXING MODEL FOR SCALAR MESONS*
Laurie M. Brown
Northwestern University, Evanston, Illinois (Received 10 March 1965)
Indirect evidence has been presented for two isoscalar mesons of spin and parity 0+: the sigma meson1 of mass mo^400 MeV and the epsilon meson2 of mass m€ «750 MeV. This situation is reminiscent of the oo-cp mixing problem and suggests that a similar model may be appropriate. It is well known that if the observed u) and cp mesons are assumed to be given by the orthogonal states
loo) = cosx |cu0> + sinA \<pQ),
I cp) = sinX I (JO0) -cosx I (p0), (1)
where o)0 and cp0 are hypothetical pure SU(3) singlet and pure SU(3) octet members, then there exists a "mixing angle" X such that the decay properties of w and cp are well explained and <pQ has a calculated mass which satisifies the Gell-Mann-Okubo relation with the other members, p and K*, of the vector octet.3 We will show that if a and e are similarly assumed to be mixtures of hypothetical pure unitary singlet and octet isosinglet members a0 and e0, the T = \ member of the octet being K(725),
then a T = l meson £ is predicted, decaying predominantly into 77 + 77. The ordering of masses in this scheme is that suggested by various bootstrap calculations.4
We shall regard as input data the masses and
widths of a and K. One additional datum is needed to completely determine the scheme and, for simplicity, we shall take for the mixing angle X the value sinx = 1/73", which will be shown to be reasonably consistent with the not very well-established properties of €.
Noting that the orthogonal matrix
/cosx sinX \ \sinx -cosX/ ' (2)
formed of the eigenvectors (1), brings the mass-square matrix
M2 (? 0 (3)
to diagonal form (where a0 and e0 are squared masses of the particles and x is a mixing mass), we obtain
A:=vT(a0-€0), a = 2ao-eo>
and
€=2€0-a0; (4)
hence, from the Gell-Mann-Okubo relation,
? = 4/c-3e0
= 4/c-2€-<x. (5)
836
VOLUME 14, NUMBER 20 P H Y S I C A L R E V I E W L E T T E R S 17 MAY 1965
E.g. , for e«/c, / | - 9 5 0 MeV; f o r e ~ p , on the other hand, / | - 8 7 5 MeV. In the la t te r case , the "equal-spacing r u l e "
K-TT=K*-P = Z-K (6)
is well sat isf ied in the form appropr ia te for an inverted octet . The order ing of levels i s in accord with predic t ions based on boots t rap calculat ions for s ca l a r mesons . 4
Turning now to the decay widths, we a s s u m e that for the four physical mesons these widths a r e de termined by S-wave decays into two pseu-dosca la r mesons , namely a-*277, e -27r , K~K + 7i, and | - 7 ] + 71. (Because of i ts even par i ty and odd G par i ty , | cannot decay s t rongly into a s ta te of fewer than five pions.) For these decay modes the decay r a t e may be wri t ten
ri = (gi2/4irm.2)(pi/2)9 (7)
where g^ is the effective coupling constant , m^ i s the m a s s of the decaying pa r t i c l e , and p^ is the momentum of one of the decay products in the r e s t f rame of the decaying pa r t i c l e . The fac tors C^=g^/4Timf a r e d imens ion less and, as will become c lear below, we t r e a t them as the cha rac t e r i s t i c constants in the spli t nonet.5
The re levant p a r t s of the SU(3) D-type couplings a r e given by
e ( T 7 r + + 7 r ' ^ = 8/3(^82/47T€0); (8a)
K+-~K° + n+, cu = 4(gQ2/4TJK); (8b)
b o
tf-ri + v0, C = 8 / 3 ( ^ - 7 4 ^ ) ; (8c) c o
or , in t e r m s of the physical pa r t i c l e s ,
(8d)
while for the singlet ,
a 0 - 7 r + + 7 r , C^ = 2 ( ^ 1 4 7 ^ ) .
For (8a), (8b), and (8d), the widths obtained with the Ce- a r e two- th i rds of the total widths of the pa r t i c l e s , and for (8c) it is the total width. These constants a r e der ived from an i n t e r a c tion Lagrangian
£f=^la0Tr(P(P+^STr<P'(p(Pf (9)
where <p and <p' a r e 3 x 3 ma t r i c e s r e p r e s e n t ing the pseudosca la r and sca l a r oc te t s , r e s p e c t ively.
Assuming again that sinA = 1/V3~, Eq. (1) leads to
C = | C + | C , a 3 d 3 a
C = i C +[K/(O + 2C)]C , (11)
where the C's in Eq. (11) al l re fe r to two- th i rds of the total widths. The total width of e for e ~P 2L€~K * s *-nen found easi ly to be about 135 MeV (taking r a = 95 MeV, rK = 23 MeV).1 '6 By cont ras t , for the case of no mixing, T € = l.SJT^, and so is about 35 MeV. In th is way one should be able to tell whether the mixing pic ture is n e c e s s a r y , a s for the vec tor mesons , o r not n e c e s s a r y , as for the pseudosca la r m e s o n s .
For the predic ted T = l meson | , decaying into 7] + 77, r^^^r with mixing and, as s tated before , i t s m a s s would be about 900 MeV. For no mixing, s ince e»/c, we would have | » / c and rt~\rK. Thus if £ is observed, i ts m a s s and width will a lso de te rmine the amount of mixing.
e 3 d 3 a (10)
*Resea rch supported by the National Science Foundation.
1 L. M. Brown and P . Singer, Phys . Rev. 133, B812 (1964); L. M. Brown and H. F a i e r , Proceedings of the Second Cora l Gables Conference on Symmetry P r i n c i p les at High E n e r g i e s , Univers i ty of Miami , January 1965 (W. H. F r e e m a n & Company, San F ranc i sco , Ca l i fornia, to be published). A bes t fit to the 7r° decay spec t r u m of the 7] meson gives m a = 400^8 MeV and r a
= 9 5 l l | MeV. 2 L. Durand, III, and Y. T. Chiu, Phys . Rev. Le t t e r s
14, 329 (1965). These au thors have found a good fit to the decay a s y m m e t r y of the p° meson by assuming m£
= 770 MeV, T e = 140 MeV. More recent ly a T = 0 s c a l a r meson called s° has been r e p o r t e d by the Univers i ty of Pennsylvania spa rk chamber group with m a s s ~720 MeV and width ~50 MeV, and conf i rmatory evidence has been seen by the Penn-Saclay bubble chamber co l laborat ion. We wish to thank P ro fe s so r W. Selove for th is information.
3In t e r m s of the "folk mode l" based upon a fundamental SU(3) t r ip le t (p,n, A), it is eas i ly shown that th i s mix ture has the p roper ty that \cp) ~ \XA) and CJ ~ 1/ ^2\pp +nn). A choice frequently made for X is sinA = 1/V3.
4Chan Hong-Mo, P . C. De Ce l l e s , and J . E. Paton, Nuovo Cimento 33, 70 (1964); B. Diu and H. R. Rubinstein, Phys . L e t t e r s £, 76 (1964); Chan Hong-Mo, Phys . Le t t e r s JJL, 269 (1964); Chan Hong-Mo and C. Wilkin, Phys . L e t t e r s 13, 263 (1964).
5If, ins tead, we were to t r e a t the g? a s independent of m a s s , we would obtain the unacceptable r e su l t that the decay r a t e into a given mode would d e c r e a s e a s we inc rea se the m a s s of the decaying pa r t i c l e .
6M. F e r r o - L u z z i et a l . , Phys . Le t t e r s 1_2, 255 (1964).
837
